1,2

(1) Row 1 of R consists of principal convergents to x.

(2) (row limits of R) = x; (column limits of R) = 0.

(3) Every positive integer occurs exactly once in D, so that as a sequence, A143516 is a permutation of the positive integers.

For any positive irrational number x, define an array D by successive rows as follows: D(n,k) = least positive integer q not already in D such that there exists an integer p such that 0 < |x - p/q| < |x - c/d| for every positive rational number c/d that has 0 < d < q. Thus p/q is the "best remaining approximate" of x when all better approximates are unavailable. For each q, define N(n,k)=p and R(n,k)=p/q. Then R is the "array of best remaining approximates of x," D is the corresponding array of denominators and N, of numerators.

Northwest corner of D:

1 2 3 5

4 6 7 10

9 11 15 19

12 14 20 27

Northwest corner of R:

2/1 3/2 5/3 8/5

6/4 10/6 11/7 16/10

15/9 18/11 24/15 31/19

19/12 23/14 32/20 44/27

Cf. A000045, A143514, A143515.

Sequence in context: A075362 A110749 A077529 this_sequence A120620 A094608 A163280

Adjacent sequences: A143513 A143514 A143515 this_sequence A143517 A143518 A143519

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Aug 22 2008